- #1

EE18

- 112

- 13

- Homework Statement
- Callen tells us the following:

Show that if a single-component system is such that ##PV^k## is constant in an

adiabatic process (k is a positive constant) the energy is given by

$$U = \frac{PV}{k-1} +Nf(\frac{PV^k}{N^k})$$

where ##f## is an arbitrary function.

Hint: ##PV^k## must be a function of S, so that ##\left( \frac{\partial U}{\partial V}

\right)_{S} = g(S)V^{-k}## where ##g(S)## is an unspecified function.

- Relevant Equations
- See my attempt below:

I was unable show that ##PV^k## must indeed equal some function of the entropy, ##g(S)##; maybe doing so would make things easier? I proceeded as below.

If we assume (as is almost surely intended by Callen) that in the given adiabatic (##d Q = 0##) process we are taking ##N## as constant and that the process is quasistatic, then we have both that ##dU = d W = -pdV## (from energy conservation/the first law) and that ##dU = TdS - pdV## (from the total differential of ##U## during this process). Thus ##TdS = 0## in such a process so that\footnote{I think we can take ##T \neq 0## as a reasonable assumption for any real process.} ##dS = 0## and thus ##S = const## during the process.

Now observe that if ##S## is held constant in a quasistatic process in which ##N## is constant, then we have ##dU = TdS - pdV +\mu dN = -pdV## while, also, ##dU = d Q -pdV## (first law for systems not changing ##N##) so that ##d Q = 0##. Thus by hypothesis, ##PV^k = const## in such a process. We'll call this constant ##C##, and note that in general it being "constant" means constant with respect to ##V##: ##C = C(S,N)## in general (why?).

Now we recall that

$$P := -\left( \frac{\partial U}{\partial V}

\right)_{S,N};$$

thus, "integrating"\footnote{See the discussion \href{https://math.stackexchange.com/ques...r-a-function-given-a-particular-partial-deriv}{here} for what this means. \label{this}} the above leads to

$$U = -\int P dV = \frac{CV^{1-k}}{k-1} + h(S,N) \stackrel{(1)}{=} \frac{PV}{k-1} + h(S,N)$$

where $h$ is some other constant function.

I can't see how to go further than this though?

Edit: I've added the following argument but am not sure why ##c## below should be invertible?

Now we note that

##u = U/N## should be zeroth order homogeneous.Thus we have that (since each term of a first order homogeneous function -- here ##U## -- must itself be fst order homogeneous) ##H## must be first-order homogeneous. Thus we see that ##H(S,N) = NH(S/N,1) \equiv Nh(s).## Analyzing ##PV^k = C(S,N)## we see that since the left-side is ##k##th order homogeneous, so too must be the right hand side. So in particular ## C(S,N) = N^kC(S/N,1) \equiv N^kc(s)##. We now argue physically that ##c## should be invertible, so that ##s = c^{-1}(Pv^k)$##so that, inserting this to the above gives

$$U = \frac{PV}{k-1} + Nh(s) = \frac{PV}{k-1} + Nh(c^{-1}(Pv^k)) \equiv \frac{PV}{k-1} + Nf(PV^k/N^k)$$

where ##f = g \circ c^{-1}##.

If we assume (as is almost surely intended by Callen) that in the given adiabatic (##d Q = 0##) process we are taking ##N## as constant and that the process is quasistatic, then we have both that ##dU = d W = -pdV## (from energy conservation/the first law) and that ##dU = TdS - pdV## (from the total differential of ##U## during this process). Thus ##TdS = 0## in such a process so that\footnote{I think we can take ##T \neq 0## as a reasonable assumption for any real process.} ##dS = 0## and thus ##S = const## during the process.

Now observe that if ##S## is held constant in a quasistatic process in which ##N## is constant, then we have ##dU = TdS - pdV +\mu dN = -pdV## while, also, ##dU = d Q -pdV## (first law for systems not changing ##N##) so that ##d Q = 0##. Thus by hypothesis, ##PV^k = const## in such a process. We'll call this constant ##C##, and note that in general it being "constant" means constant with respect to ##V##: ##C = C(S,N)## in general (why?).

Now we recall that

$$P := -\left( \frac{\partial U}{\partial V}

\right)_{S,N};$$

thus, "integrating"\footnote{See the discussion \href{https://math.stackexchange.com/ques...r-a-function-given-a-particular-partial-deriv}{here} for what this means. \label{this}} the above leads to

$$U = -\int P dV = \frac{CV^{1-k}}{k-1} + h(S,N) \stackrel{(1)}{=} \frac{PV}{k-1} + h(S,N)$$

where $h$ is some other constant function.

I can't see how to go further than this though?

Edit: I've added the following argument but am not sure why ##c## below should be invertible?

Now we note that

##u = U/N## should be zeroth order homogeneous.Thus we have that (since each term of a first order homogeneous function -- here ##U## -- must itself be fst order homogeneous) ##H## must be first-order homogeneous. Thus we see that ##H(S,N) = NH(S/N,1) \equiv Nh(s).## Analyzing ##PV^k = C(S,N)## we see that since the left-side is ##k##th order homogeneous, so too must be the right hand side. So in particular ## C(S,N) = N^kC(S/N,1) \equiv N^kc(s)##. We now argue physically that ##c## should be invertible, so that ##s = c^{-1}(Pv^k)$##so that, inserting this to the above gives

$$U = \frac{PV}{k-1} + Nh(s) = \frac{PV}{k-1} + Nh(c^{-1}(Pv^k)) \equiv \frac{PV}{k-1} + Nf(PV^k/N^k)$$

where ##f = g \circ c^{-1}##.

Last edited: